a. John called with a message for you to call your Mom.
b. John called with a message that you should call your Mom.

c. John called with a message to call your Mom.

d. The message was for you to call your Mom.
e. The message was that you should call your Mom.[

f. The message was to call your Mom.

Are the above sentences grammatical?

Sentence (c) is supposed to mean the same as (a) and (b). I suppose that technically speaking it could have other meanings since it is not clear who is to call the Mom in question. Maybe the message was for the speaker.

Sentence (f) is supposed to mean the same as (d) and (e), but as far as I can see it has the same problem as (c).

There are times when the use of quotation-marks helps to clarify; f. might read, The message was to "call your Mom".
The punctuation may help the reader to understand it as, The message was [for you] to "call your Mom".
Technically, the sentence, if it uses quotation-marks, would include the quote verbatim: Your Mom called, saying "tell Tisnelda to call me, please".

They all seem to me to be normal and idiomatic. For me, "idiomatic" trumps "grammatical" every time, although I can't see any grammatical issues with any of them. Many idiomatic expressions are elliptical, and sentences c and f are elliptical, but not ungrammatical.

Essentially, ellipsis in grammar is when we leave words out of an utterance, although they may be required for grammatical completeness. The first sentence of this post shows a very common ellipsis (the subject "it" is missed out: "It could take ...").

|f you look for them, you will find many examples in normal speech.

"I should really be grateful, but I'm not." (I'm not grateful.)

Indeed, in the example above, we would often simply say "I should be grateful." It could be argued that the entire "but I am not grateful" is ellipted.

Another very common example is the loss of a relative pronoun and verb in relative clauses: "The man walking his dog on the beach". ("The man who was walking his dog on the beach."

Another common one in speech, but not so much in writing, is the loss of an auxiliary verb and subject in questions: "Going to the match tonight?".

Personally, I am very fond of explaining some odd syntactical constructions, and even some difficult semantic relationships in terms of ellipsis. It makes life much easier.

The other use of the word "ellipsis", of course, is to refer to the three dots that indicate that some words have been omitted in a text.

The above was written in rather a hurry. It is less coherent than I would have liked.

The important thing about ellipsis is that, although one or more words are missed out from a sentence, thus rendering it grammatically incomplete in a strictly logical sense, the meaning is clear. In some cases, this is done on an ad hoc basis, but as often as not elliptical expressions have become fixed idiomatic usages:

The bigger the better
(two verbs, and two nouns are needed for grammatical completeness)

Hitler has one ball, Goebbels none
(The sentence requires another verb for grammatical completeness - and three balls for anatomical completeness)

Fire when ready
(In this fixed idiom, the "you are" is missing)

As I say, ellipsis can be invoked as a way of explaining some otherwise difficult grammatical issues. I remember many years ago attempting a grammatical analysis of "we are going fishing tomorrow" in which I argued that the construction "go fishing" is in fact elliptical. I'm not sure I would use the same argument nowadays, but it is probably a valid option.